Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.7% → 88.0%

Time: 16.3s

Alternatives: 5

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 88.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\left(0.5 \cdot \left(D\_m \cdot \frac{M}{d}\right)\right) \cdot \sqrt{h}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (if (<= h -5e-310)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* M (* 0.5 (/ D_m d))) 2.0)) l))))
   (* w0 (sqrt (- 1.0 (/ (pow (* (* 0.5 (* D_m (/ M d))) (sqrt h)) 2.0) l))))))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (h <= -5e-310) {
		tmp = w0 * sqrt((1.0 - ((h * pow((M * (0.5 * (D_m / d))), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (pow(((0.5 * (D_m * (M / d))) * sqrt(h)), 2.0) / l)));
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (h <= (-5d-310)) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((m * (0.5d0 * (d_m / d))) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - ((((0.5d0 * (d_m * (m / d))) * sqrt(h)) ** 2.0d0) / l)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (h <= -5e-310) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((M * (0.5 * (D_m / d))), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (Math.pow(((0.5 * (D_m * (M / d))) * Math.sqrt(h)), 2.0) / l)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	tmp = 0
	if h <= -5e-310:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((M * (0.5 * (D_m / d))), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (math.pow(((0.5 * (D_m * (M / d))) * math.sqrt(h)), 2.0) / l)))
	return tmp

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(M * Float64(0.5 * Float64(D_m / d))) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(0.5 * Float64(D_m * Float64(M / d))) * sqrt(h)) ^ 2.0) / l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp_2 = code(w0, M, D_m, h, l, d)
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = w0 * sqrt((1.0 - ((h * ((M * (0.5 * (D_m / d))) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 - ((((0.5 * (D_m * (M / d))) * sqrt(h)) ^ 2.0) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := If[LessEqual[h, -5e-310], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(0.5 * N[(D$95$m * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 79.5%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   3. Simplified 81.2%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 84.5%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}} \]

      2. add-sqr-sqrt 84.5%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\sqrt{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}} \cdot \sqrt{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}}\right)} \cdot h}{\ell}} \]

      3. pow2 84.5%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{{\left(\sqrt{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}}\right)}^{2}} \cdot h}{\ell}} \]

      4. unpow2 84.5%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\sqrt{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right) \cdot \left(M \cdot \frac{D}{2 \cdot d}\right)}}\right)}^{2} \cdot h}{\ell}} \]

      5. sqrt-prod 46.3%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(\sqrt{M \cdot \frac{D}{2 \cdot d}} \cdot \sqrt{M \cdot \frac{D}{2 \cdot d}}\right)}}^{2} \cdot h}{\ell}} \]

      6. add-sqr-sqrt 84.5%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}} \]

      7. *-un-lft-identity 84.5%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \frac{\color{blue}{1 \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}} \]

      8. times-frac 84.5%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}\right)}^{2} \cdot h}{\ell}} \]

      9. metadata-eval 84.5%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}} \]

   6. Applied egg-rr 84.5%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}} \]

   if -4.999999999999985e-310 < h

   1. Initial program 79.4%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   3. Simplified 77.4%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 85.3%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}} \]

      2. add-sqr-sqrt 85.3%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\sqrt{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}} \cdot \sqrt{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}}\right)} \cdot h}{\ell}} \]

      3. pow2 85.3%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{{\left(\sqrt{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}}\right)}^{2}} \cdot h}{\ell}} \]

      4. unpow2 85.3%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\sqrt{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right) \cdot \left(M \cdot \frac{D}{2 \cdot d}\right)}}\right)}^{2} \cdot h}{\ell}} \]

      5. sqrt-prod 53.0%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(\sqrt{M \cdot \frac{D}{2 \cdot d}} \cdot \sqrt{M \cdot \frac{D}{2 \cdot d}}\right)}}^{2} \cdot h}{\ell}} \]

      6. add-sqr-sqrt 85.3%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot h}{\ell}} \]

      7. *-un-lft-identity 85.3%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \frac{\color{blue}{1 \cdot D}}{2 \cdot d}\right)}^{2} \cdot h}{\ell}} \]

      8. times-frac 85.3%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}\right)}^{2} \cdot h}{\ell}} \]

      9. metadata-eval 85.3%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}} \]

   6. Applied egg-rr 85.3%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 85.3%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\sqrt{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h} \cdot \sqrt{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}}}{\ell}} \]

      2. pow2 85.3%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{{\left(\sqrt{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}\right)}^{2}}}{\ell}} \]

      3. sqrt-prod 85.3%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(\sqrt{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}} \cdot \sqrt{h}\right)}}^{2}}{\ell}} \]

      4. unpow2 85.3%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\sqrt{\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}} \cdot \sqrt{h}\right)}^{2}}{\ell}} \]

      5. sqrt-prod 55.8%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\color{blue}{\left(\sqrt{M \cdot \left(0.5 \cdot \frac{D}{d}\right)} \cdot \sqrt{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}\right)} \cdot \sqrt{h}\right)}^{2}}{\ell}} \]

      6. add-sqr-sqrt 88.7%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)} \cdot \sqrt{h}\right)}^{2}}{\ell}} \]

      7. associate-*r/ 88.7%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\left(M \cdot \color{blue}{\frac{0.5 \cdot D}{d}}\right) \cdot \sqrt{h}\right)}^{2}}{\ell}} \]

   8. Applied egg-rr 88.7%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{{\left(\left(M \cdot \frac{0.5 \cdot D}{d}\right) \cdot \sqrt{h}\right)}^{2}}}{\ell}} \]

   9. Taylor expanded in M around 0 90.7%

      \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\color{blue}{\left(0.5 \cdot \frac{D \cdot M}{d}\right)} \cdot \sqrt{h}\right)}^{2}}{\ell}} \]
   10. Step-by-step derivation

       1. associate-/l* 89.9%

          \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\left(0.5 \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right) \cdot \sqrt{h}\right)}^{2}}{\ell}} \]

   11. Simplified 89.9%

       \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\color{blue}{\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)} \cdot \sqrt{h}\right)}^{2}}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \sqrt{h}\right)}^{2}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D\_m}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -200:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \left(0.5 \cdot D\_m\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D_m) (* d 2.0)) 2.0) (/ h l)) -200.0)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M d) (* 0.5 D_m)) 2.0)))))
   w0))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D_m) / (d * 2.0)), 2.0) * (h / l)) <= -200.0) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) * (0.5 * D_m)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((m * d_m) / (d * 2.0d0)) ** 2.0d0) * (h / l)) <= (-200.0d0)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((m / d) * (0.5d0 * d_m)) ** 2.0d0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D_m) / (d * 2.0)), 2.0) * (h / l)) <= -200.0) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) * (0.5 * D_m)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M * D_m) / (d * 2.0)), 2.0) * (h / l)) <= -200.0:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) * (0.5 * D_m)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D_m) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)) <= -200.0)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / d) * Float64(0.5 * D_m)) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp_2 = code(w0, M, D_m, h, l, d)
	tmp = 0.0;
	if (((((M * D_m) / (d * 2.0)) ^ 2.0) * (h / l)) <= -200.0)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) * (0.5 * D_m)) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -200.0], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / d), $MachinePrecision] * N[(0.5 * D$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -200

   1. Initial program 65.6%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 65.6%

         \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot \frac{h}{\ell}} \]

      2. times-frac 66.8%

         \[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{d} \cdot \frac{D}{2}\right)}}^{2} \cdot \frac{h}{\ell}} \]

      3. div-inv 66.8%

         \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M}{d} \cdot \color{blue}{\left(D \cdot \frac{1}{2}\right)}\right)}^{2} \cdot \frac{h}{\ell}} \]

      4. metadata-eval 66.8%

         \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M}{d} \cdot \left(D \cdot \color{blue}{0.5}\right)\right)}^{2} \cdot \frac{h}{\ell}} \]

   4. Applied egg-rr 66.8%

      \[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}}^{2} \cdot \frac{h}{\ell}} \]

   if -200 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 85.8%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   3. Simplified 86.4%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 95.6%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq -200:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \left(0.5 \cdot D\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-198}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(M \cdot \frac{D\_m}{d \cdot 2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (if (<= (/ h l) -5e-198)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* M (/ D_m (* d 2.0))) 2.0)))))
   w0))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -5e-198) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((M * (D_m / (d * 2.0))), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((h / l) <= (-5d-198)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((m * (d_m / (d * 2.0d0))) ** 2.0d0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -5e-198) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((M * (D_m / (d * 2.0))), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	tmp = 0
	if (h / l) <= -5e-198:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((M * (D_m / (d * 2.0))), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -5e-198)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(M * Float64(D_m / Float64(d * 2.0))) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp_2 = code(w0, M, D_m, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -5e-198)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((M * (D_m / (d * 2.0))) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -5e-198], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(D$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -4.9999999999999999e-198

   1. Initial program 76.8%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   3. Simplified 74.9%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
   4. Add Preprocessing

   if -4.9999999999999999e-198 < (/.f64 h l)

   1. Initial program 82.3%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   3. Simplified 83.9%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 93.6%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-198}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.02 \cdot 10^{-84}:\\ \;\;\;\;w0 \cdot \left(1 + \left({D\_m}^{2} \cdot \left(h \cdot \left(\frac{M}{d} \cdot \frac{\frac{M}{d}}{\ell}\right)\right)\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (if (<= d 1.02e-84)
   (* w0 (+ 1.0 (* (* (pow D_m 2.0) (* h (* (/ M d) (/ (/ M d) l)))) -0.125)))
   w0))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 1.02e-84) {
		tmp = w0 * (1.0 + ((pow(D_m, 2.0) * (h * ((M / d) * ((M / d) / l)))) * -0.125));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= 1.02d-84) then
        tmp = w0 * (1.0d0 + (((d_m ** 2.0d0) * (h * ((m / d) * ((m / d) / l)))) * (-0.125d0)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double tmp;
	if (d <= 1.02e-84) {
		tmp = w0 * (1.0 + ((Math.pow(D_m, 2.0) * (h * ((M / d) * ((M / d) / l)))) * -0.125));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	tmp = 0
	if d <= 1.02e-84:
		tmp = w0 * (1.0 + ((math.pow(D_m, 2.0) * (h * ((M / d) * ((M / d) / l)))) * -0.125))
	else:
		tmp = w0
	return tmp

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	tmp = 0.0
	if (d <= 1.02e-84)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64((D_m ^ 2.0) * Float64(h * Float64(Float64(M / d) * Float64(Float64(M / d) / l)))) * -0.125)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp_2 = code(w0, M, D_m, h, l, d)
	tmp = 0.0;
	if (d <= 1.02e-84)
		tmp = w0 * (1.0 + (((D_m ^ 2.0) * (h * ((M / d) * ((M / d) / l)))) * -0.125));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := If[LessEqual[d, 1.02e-84], N[(w0 * N[(1.0 + N[(N[(N[Power[D$95$m, 2.0], $MachinePrecision] * N[(h * N[(N[(M / d), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 1.02000000000000004e-84

   1. Initial program 79.8%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   3. Simplified 79.2%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 50.7%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 50.7%

         \[\leadsto w0 \cdot \left(1 + \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot -0.125}\right) \]

      2. associate-/l* 50.8%

         \[\leadsto w0 \cdot \left(1 + \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)} \cdot -0.125\right) \]

   7. Simplified 50.8%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot -0.125\right)} \]
   8. Step-by-step derivation

      1. *-commutative 50.8%

         \[\leadsto w0 \cdot \left(1 + \left({D}^{2} \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{{d}^{2} \cdot \ell}\right) \cdot -0.125\right) \]

      2. associate-*r/ 50.9%

         \[\leadsto w0 \cdot \left(1 + \left({D}^{2} \cdot \color{blue}{\left(h \cdot \frac{{M}^{2}}{{d}^{2} \cdot \ell}\right)}\right) \cdot -0.125\right) \]

      3. *-commutative 50.9%

         \[\leadsto w0 \cdot \left(1 + \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2}}{{d}^{2} \cdot \ell} \cdot h\right)}\right) \cdot -0.125\right) \]

      4. associate-/r* 53.9%

         \[\leadsto w0 \cdot \left(1 + \left({D}^{2} \cdot \left(\color{blue}{\frac{\frac{{M}^{2}}{{d}^{2}}}{\ell}} \cdot h\right)\right) \cdot -0.125\right) \]

      5. unpow2 53.9%

         \[\leadsto w0 \cdot \left(1 + \left({D}^{2} \cdot \left(\frac{\frac{\color{blue}{M \cdot M}}{{d}^{2}}}{\ell} \cdot h\right)\right) \cdot -0.125\right) \]

      6. unpow2 53.9%

         \[\leadsto w0 \cdot \left(1 + \left({D}^{2} \cdot \left(\frac{\frac{M \cdot M}{\color{blue}{d \cdot d}}}{\ell} \cdot h\right)\right) \cdot -0.125\right) \]

      7. frac-times 63.0%

         \[\leadsto w0 \cdot \left(1 + \left({D}^{2} \cdot \left(\frac{\color{blue}{\frac{M}{d} \cdot \frac{M}{d}}}{\ell} \cdot h\right)\right) \cdot -0.125\right) \]

      8. pow2 63.0%

         \[\leadsto w0 \cdot \left(1 + \left({D}^{2} \cdot \left(\frac{\color{blue}{{\left(\frac{M}{d}\right)}^{2}}}{\ell} \cdot h\right)\right) \cdot -0.125\right) \]

   9. Applied egg-rr 63.0%

      \[\leadsto w0 \cdot \left(1 + \left({D}^{2} \cdot \color{blue}{\left(\frac{{\left(\frac{M}{d}\right)}^{2}}{\ell} \cdot h\right)}\right) \cdot -0.125\right) \]
   10. Step-by-step derivation

       1. unpow2 63.0%

          \[\leadsto w0 \cdot \left(1 + \left({D}^{2} \cdot \left(\frac{\color{blue}{\frac{M}{d} \cdot \frac{M}{d}}}{\ell} \cdot h\right)\right) \cdot -0.125\right) \]

   11. Applied egg-rr 63.0%

       \[\leadsto w0 \cdot \left(1 + \left({D}^{2} \cdot \left(\frac{\color{blue}{\frac{M}{d} \cdot \frac{M}{d}}}{\ell} \cdot h\right)\right) \cdot -0.125\right) \]
   12. Step-by-step derivation

       1. associate-/l* 64.1%

          \[\leadsto w0 \cdot \left(1 + \left({D}^{2} \cdot \left(\color{blue}{\left(\frac{M}{d} \cdot \frac{\frac{M}{d}}{\ell}\right)} \cdot h\right)\right) \cdot -0.125\right) \]

   13. Applied egg-rr 64.1%

       \[\leadsto w0 \cdot \left(1 + \left({D}^{2} \cdot \left(\color{blue}{\left(\frac{M}{d} \cdot \frac{\frac{M}{d}}{\ell}\right)} \cdot h\right)\right) \cdot -0.125\right) \]

   if 1.02000000000000004e-84 < d

   1. Initial program 78.9%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   3. Simplified 79.2%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 77.8%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.02 \cdot 10^{-84}:\\ \;\;\;\;w0 \cdot \left(1 + \left({D}^{2} \cdot \left(h \cdot \left(\frac{M}{d} \cdot \frac{\frac{M}{d}}{\ell}\right)\right)\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.5% accurate, 216.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ w0 \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d) :precision binary64 w0)

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	return w0;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    code = w0
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	return w0;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	return w0

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	return w0
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp = code(w0, M, D_m, h, l, d)
	tmp = w0;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := w0

Derivation

1. Initial program 79.5%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

3. Simplified 79.2%

   \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
4. Add Preprocessing

5. Taylor expanded in M around 0 67.8%

   \[\leadsto \color{blue}{w0} \]

6. Final simplification 67.8%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))